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Properties

Label 16.16.1027753048...5056.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{24}\cdot 3^{4}\cdot 229^{8}$
Root discriminant $56.33$
Ramified primes $2, 3, 229$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $\GL(2,Z/4)$ (as 16T193)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-51, -414, -182, 4634, 6358, -14072, -22354, 8398, 17908, -1696, -5614, 174, 783, -8, -48, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 48*x^14 - 8*x^13 + 783*x^12 + 174*x^11 - 5614*x^10 - 1696*x^9 + 17908*x^8 + 8398*x^7 - 22354*x^6 - 14072*x^5 + 6358*x^4 + 4634*x^3 - 182*x^2 - 414*x - 51)
 
gp: K = bnfinit(x^16 - 48*x^14 - 8*x^13 + 783*x^12 + 174*x^11 - 5614*x^10 - 1696*x^9 + 17908*x^8 + 8398*x^7 - 22354*x^6 - 14072*x^5 + 6358*x^4 + 4634*x^3 - 182*x^2 - 414*x - 51, 1)
 

Normalized defining polynomial

\( x^{16} - 48 x^{14} - 8 x^{13} + 783 x^{12} + 174 x^{11} - 5614 x^{10} - 1696 x^{9} + 17908 x^{8} + 8398 x^{7} - 22354 x^{6} - 14072 x^{5} + 6358 x^{4} + 4634 x^{3} - 182 x^{2} - 414 x - 51 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(10277530482246004726673965056=2^{24}\cdot 3^{4}\cdot 229^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.33$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 229$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{15} a^{13} + \frac{1}{15} a^{12} + \frac{1}{15} a^{11} - \frac{2}{5} a^{10} + \frac{1}{5} a^{9} - \frac{1}{5} a^{8} + \frac{4}{15} a^{7} + \frac{1}{5} a^{6} + \frac{4}{15} a^{5} + \frac{1}{5} a^{4} - \frac{1}{3} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{225} a^{14} - \frac{1}{225} a^{13} + \frac{1}{25} a^{12} - \frac{73}{225} a^{11} + \frac{2}{45} a^{10} - \frac{11}{25} a^{9} - \frac{1}{45} a^{8} - \frac{7}{45} a^{7} - \frac{112}{225} a^{6} + \frac{1}{9} a^{5} - \frac{76}{225} a^{4} + \frac{4}{225} a^{3} + \frac{73}{225} a^{2} + \frac{4}{15} a + \frac{7}{75}$, $\frac{1}{51859273555371731625} a^{15} + \frac{31575179642538694}{17286424518457243875} a^{14} - \frac{11761327217266109}{4714479414124702875} a^{13} - \frac{5785178682748458826}{51859273555371731625} a^{12} + \frac{7746905673024355892}{17286424518457243875} a^{11} + \frac{10914436915098410356}{51859273555371731625} a^{10} + \frac{749044931379048328}{51859273555371731625} a^{9} + \frac{519509740248321908}{2074370942214869265} a^{8} - \frac{1390020189315374689}{17286424518457243875} a^{7} - \frac{2141922781076113907}{17286424518457243875} a^{6} + \frac{5744890862837423008}{17286424518457243875} a^{5} - \frac{397971092639710706}{5762141506152414625} a^{4} + \frac{336195489106239616}{10371854711074346325} a^{3} - \frac{21164686931952499181}{51859273555371731625} a^{2} + \frac{1406592348763044164}{5762141506152414625} a - \frac{468147921030634919}{17286424518457243875}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{3}$, which has order $3$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 548711917.38 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$\GL(2,Z/4)$ (as 16T193):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 96
The 14 conjugacy class representatives for $\GL(2,Z/4)$
Character table for $\GL(2,Z/4)$

Intermediate fields

\(\Q(\sqrt{229}) \), 4.4.14656.1, 4.4.2517168.1, 8.8.11264239538176.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 12 siblings: 12.12.1880463412742639616.1, 12.12.7521853650970558464.1, 12.12.996819725736260352.1, 12.12.1880463412742639616.2
Degree 16 sibling: 16.16.15874601343641928698167296.1
Degree 24 siblings: data not computed
Degree 32 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
229Data not computed